Weak injectivity and auxiliary-pattern reduction

Apply [F13] to the finite compact input for the branch at hand. For TOP the ambient product or ball is already standard; for the exotic branch its actual marked product/ball homeomorphism is given and is encoded in both directions. Use two coding passes. First encode only this finite compact geometry by a real $r_0$. In $L[r_0]$, before imposing any localized homology or strong-winding condition, use the ordinary finite smooth Morse construction to obtain finite handle/CW presentations of the supplied smooth ambient manifolds and finite presentations of the relevant knot exteriors, with their comparison maps, homotopies and peripheral markings. Export these purely geometric presentations and compact comparison certificates by [F13]. No choice-free generic-Morse theorem in $V$ is being assumed. Now work in $V$: its given localized homology hypothesis applies to these actual exported presentations. Over the PID $R_n$, the augmented finite free chain complex for an $R_n$-acyclic ball, or the mapping cone for either boundary inclusion, has a finite matrix contracting homotopy; finite row/column reduction supplies it. Likewise, the given strong-winding condition supplies, for each generator in the finite closure-knot presentation, an actual finite word expressing it as a product of conjugates of the designated axis meridian. There are only finitely many such generators and certificates, so choosing this list needs only finite choice. Enlarge $r_0$ to one real $r$ containing the presentations, their actual comparison certificates, these input group words and the finite contraction matrices. Their displayed equations $dh+hd=1$ and word identities are arithmetic, and establish the exact input homology and strong-winding hypotheses inside $L[r]$. No raw homology or fundamental-group absoluteness is asserted. In that model apply the unconditional CDR weak geometry, including [F11], [F12] and the finite connected-sum identities needed in [F4]. Export the resulting compact manifolds, disks, homeomorphisms and finite homology certificates using [F13]. Both stages carry finite normal-generation certificates: the input axis words just described accompany the final CDR call, and the output cobordisms are exported with explicit words expressing each generator in their finite presentations as a product of conjugates of the designated boundary generators. These are actual finite witnesses in $V$. The only ambient hypotheses in these applications are the actual coded ones, and no family of annuli or arbitrary points is selected. In the ordinary smooth branch export only the unconditionally constructed contractible smooth $B$ and its marked co-core; do not assume the ambient smooth Poincare hypothesis in $L[r]$. After decoding in $V$, cap the actual $B$ there, and only then use the supplied smooth Poincare hypothesis in the subsequent smooth-branch argument.

Now suppose P(K)=P(J). By [F6], the connected-sum identity and the ribbon inverse give $P(J)=P_K(-K\mathrm{\#}J)$ as concordance classes. Define R as the local connected sum of −P(K) with P_K. Then R(−K#J) is concordant to U because K#(−K#J) is concordant to J and satellites preserve concordance; moreover R(U)=−P(K)#P(K). By [F10], this closure is ribbon; [F6] gives w(R)=w(P).

Suppose P(K)=P(U), with the stated closure-slice hypothesis. By [F7] this equality is witnessed by a concordance annulus; the zero-surgery descent [F12], with the retained normal-GIVEN construction [F8], gives a meridian-preserving homology cobordism from M(P(K)) to M(P(U)). Applying [F2] to K and U gives cobordisms from M(P(K)) to M(K) and from M(P(U)) to M(U). On each pattern cobordism its marked first-homology relation is [mu_companion]=n[mu_satellite], where n is a unit in R_n; the concordance cobordism from [F8] identifies the two satellite meridian classes exactly. No equality of the pattern and companion preferred meridians is assumed.

Glue the cobordisms along the identified surgery manifolds. Mayer–Vietoris shows that the resulting boundary inclusions retain the required homology isomorphisms. In the exotic/topological branches, van Kampen shows the boundary normal-generation condition is preserved under this gluing. Hence M(K) and M(U) satisfy the corresponding surgery-cobordism relation. In the localized branch each endpoint meridian maps to a unit multiple of the same H_1 generator: compose the two winding-unit relations and the intervening exact concordance relation. In particular each endpoint meridian, including that of K, remains an R_n-module generator. In the strong-winding branches n=+1 or -1, so these multipliers are already integral units.

In the localized category, cap the M(U)=S¹×S² end with S¹×B³. Mayer–Vietoris gives an R_n-homology circle X with boundary M(K), and its meridian class is an R_n-module generator by step 3.1. Attach a smooth 2-handle along that meridian with the surgery-dual zero framing. The relative pair (B,X) has only H_2(B,X;R_n)=R_n, generated by its core; its boundary map to H_1(X;R_n)=R_n is the meridian class, hence multiplication by a unit and an isomorphism. The pair sequence gives H_i(B;R_n)=0 for all i>0 and H_0(B;R_n)=R_n. The boundary handle description is the zero-surgery presentation for K together with its zero-framed meridian; this dual meridian surgery restores S³, and the 2-handle co-core has boundary K under that identification. Thus the resulting B is an R_n-homology ball containing the required smooth proper disk, proving K=U in C^{1/n}. Only a unit meridian multiplier was used, so the corrected [F2] supplies exactly the needed premise.

In the exotic and topological categories, the normal-generation condition makes the capped manifold after the meridian 2-handle contractible with boundary S³. The external input [F11] identifies this manifold with the boundary-marked topological $B^4$. Its co-core is smooth in the exotic branch and locally flat in the TOP branch, and its boundary is the original marked $K$. It therefore gives a pseudo-slice disk or a TOP slice disk respectively, proving $K=U$ in the required category.

In the ordinary smooth branch, work now in the ambient universe $V$: if P(U) is smoothly slice and P has strong winding ±1, the same cap and handle construction gives a contractible smooth 4-manifold B with boundary S³. Capping B with standard B⁴ gives a smooth homotopy S⁴; under the smooth 4-dimensional Poincaré conjecture and [F5], B is diffeomorphic to B⁴.

If P has strong winding ±1, [F9] implies R also has strong winding ±1. The closure R(U) supplies the localized slice, pseudo-slice, topological-slice, and standard smooth-slice hypotheses in their respective branches. Apply the applicable earlier weak-injectivity branch to R to obtain $-K\mathrm{\#}J=U$ and hence $K=J$.